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                <h1 id="写在前面"><a href="#写在前面" class="headerlink" title="写在前面"></a>写在前面</h1><p>距离上次写学习笔记差不多已经一年半了，可恶，还差一年鼠鼠我就可以当练习生了耶😭</p>
<p>直到上了研究生才发现和本科区别不是很大，可能有还是研一的缘由，还有可能是我导师太忙管不到我的缘由，大部分知识还是要自己学的，目前已知的信息就是未来的两大研究方向都和机器学习有关，所以咱们博客复出的开篇就是机器学习入门系列，希望研究生的两年半能很好的提升自己吧😁。</p>
<span id="more"></span>
<h1 id="ML的三大步骤"><a href="#ML的三大步骤" class="headerlink" title="ML的三大步骤"></a>ML的三大步骤</h1><p>我们通过预测宝可梦进化后的cp值举例。</p>
<h2 id="1-确定Model"><a href="#1-确定Model" class="headerlink" title="1.确定Model"></a>1.确定Model</h2><p>即定义一个函数f，使得输出出来的结果和真正的结果近似最好是相等。这里并没有明确的思路，只能凭经验去尝试，我们从最简单的线性模型开始。</p>
<script type="math/tex; mode=display">
y = b + w*Xcp (其中Xcp是宝可梦进化前的cp)</script><p>w和b是未知值，可以取任意值，而上述式子抽象出来即叫做Model。</p>
<p>上面的式子只考虑了进化前的cp值，而进化前我们还有很多因素可以考虑，因此我们可以写一个更复杂的线性模型：</p>
<script type="math/tex; mode=display">
y = b + \Sigma w_i * X_i</script><p>参数描述：y——进化后cp预测值；b：bais；Xi：进化前的属性；wi：属性的权重</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923121104153.png" alt="image-20220923121104153" style="zoom:80%;"></p>
<h2 id="2-Define-Loss"><a href="#2-Define-Loss" class="headerlink" title="2.Define Loss"></a>2.Define Loss</h2><p>我们确定好model后需要定义一个评估函数来决定模型的好坏，通常称之为Loss Function，在这个例子里，我们可以这样写：</p>
<script type="math/tex; mode=display">
L(f) = L(w,b)</script><p>因为我们的模型输出y是由w和b决定的，因此Loss函数的input也是w和b，也就是说，Loss function其实是在评估一组参数的好坏。</p>
<p>参数说明：</p>
<script type="math/tex; mode=display">
X^i:表示第i只宝可梦\\
\widehat{Y}^i：表示第i只宝可梦实际的cp值</script><p>Loss Function和Model一样也是我们自己来定义的，通常我们用方差的形式来衡量，即预测值和真值之间的差的平方和</p>
<script type="math/tex; mode=display">
L(f) = L(w,b) = \Sigma(\widehat{y}^n-y^n) = \Sigma(\widehat{y}^n-(b+w_n*Xcp^n))</script><p>由式子我们可以看出Loss越大，说明参数越不好。</p>
<p>我们可以通过Loss function的可视化来清晰的选出最好的w和b。下图(b,w)的每个点的像素颜色代表loss的大小，越红loss越大，越蓝越小，我们可以看出画叉的那个点的(b,w)是Loss最小的。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923122809961.png" alt="image-20220923122809961" style="zoom:80%;"></p>
<h2 id="3-优化"><a href="#3-优化" class="headerlink" title="3.优化"></a>3.优化</h2><p>第二步做完，我们已经确定好了我们的Loss function，接下来就是优化找到一个最好的模型function，其实就是找到使得L(f)=L(w,b)最小的那个w<em> 和 b</em>。</p>
<p>这里介绍一种普遍的方法——gradient descent(梯度下降法)。</p>
<h3 id="Gradient-descent"><a href="#Gradient-descent" class="headerlink" title="Gradient descent"></a>Gradient descent</h3><p>gradient descent的厉害之处在于，只要L(f)是可微分的，gradient descent都可以拿来处理这个f，找到表现比较好的parameters。</p>
<p>我们举个单参数的例子L(w)，我们的目标就是要找到使得L最小的参数w*。</p>
<p>梯度下降法分为以下几步：</p>
<ol>
<li><p>首先随机选取一个点wo</p>
</li>
<li><p>计算函数在该点的微分，若是负的，说明左大右小，应使得w变大；；若是正的，应使得w变小</p>
</li>
<li><p>w的改变量step size不止决定于该点的微分，还决定于一个超参数η，我们成为learning rate，如果η越大，那么w每次移动的步幅也就越大，反之也越小。<br>如果learning rate设置的大一点，那么也就代表着机器学习的速度快一点，但是有可能导致步子迈大了错过了最合适的全局最小的点。</p>
</li>
<li><script type="math/tex; mode=display">
w1 = w0 -η*\frac{dL}{dw}|w=w0\\
w2 = w1 -η*\frac{dL}{dw}|w=w1\\
……\\</script></li>
</ol>
<p>当斜率为0时，即找到一个极小值，但是我们观察下图，会发现有找到极小值但是不是全局最小值的现象。这种问题我们在线性模型里其实并不用担心，因为在线性模型中我们的Loss function其实是个convex，就是凸函数，他只有一个global minima全局最优解。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923123706200.png" alt="image-20220923123706200" style="zoom:80%;"></p>
<h3 id="评估系统"><a href="#评估系统" class="headerlink" title="评估系统"></a>评估系统</h3><p>回到我们的例子来，我们在loss function中定义了两个参数L(w,b)，其实本质上和单参数问题是一样的解决方法，只不过微分变味了偏微分。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923125658340.png" alt="image-20220923125658340"></p>
<p>用梯度下降法我们可以得到最好的参数是b=-188.4, w=2.7</p>
<p>我们需要一套评估系统来观测我们预测的值和实际值的误差error的大小，这里又分为训练数据和测试数据的误差。</p>
<p>training data里得到的误差一般是要比testing data要小，这也符合常识，当然我们实际关心的就是测试数据的数据（训练没见过的数据）。</p>
<h3 id="重新设计模型"><a href="#重新设计模型" class="headerlink" title="重新设计模型"></a>重新设计模型</h3><p>实际上，从结果来看，最终的function可能不是一条直线，可能是稍微更复杂一点的曲线，我们考虑二次方的model，即</p>
<script type="math/tex; mode=display">
y = b + w1*Xcp+w2*(Xcp)^2</script><p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923130734911.png" alt="image-20220923130734911" style="zoom:80%;"></p>
<p>考虑三次方的model：</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923130754280.png" alt="image-20220923130754280"></p>
<p>考虑四次方的model：</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923130849744.png" alt="image-20220923130849744" style="zoom:80%;"></p>
<p>考虑五次方：</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923130911914.png" alt="image-20220923130911914" style="zoom:80%;"></p>
<p>我们由上述五个model的表现可以观测出：</p>
<p>随着高次项的增加，训练数据的average error会逐渐减少，但是这不是我们关心的，我们发现在测试数据上，model复杂到一定程度后，error不减反增，这种现象，我们称之为过拟合overfitting。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923131213015.png" alt="image-20220923131213015" style="zoom:80%;"></p>
<p>因此不是model越复杂越好，我们需要选一个最合适的model。</p>
<h3 id="引入更多参数"><a href="#引入更多参数" class="headerlink" title="引入更多参数"></a>引入更多参数</h3><p>我们收集更多数据放入training data，发现进化前后的cp值还与种族有关联，记种族为Xs。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923135122803.png" alt="image-20220923135122803" style="zoom:80%;"></p>
<p>重新设计model：<img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923135347083.png" alt="image-20220923135347083" style="zoom:80%;"></p>
<p>这里引入δ(条件表达式)的概念，当条件表达式为true，则δ为1；当条件表达式为false，则δ为0，因此可以通过下图的方式，将4个if语句转化成同一个linear model</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923135457401.png" alt="image-20220923135457401" style="zoom:80%;"></p>
<p>我们发现引入种族Xs后，training error和testing error都明显下降：</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923135601853.png" alt="image-20220923135601853" style="zoom:80%;"></p>
<p>接着我们观测是不是参数考虑得越多越好，我们将可能的参数都引入，设计出一个最复杂的model：</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923135820911.png" alt="image-20220923135820911" style="zoom:80%;"></p>
<p>我们发现模型设计的复杂反而更容易造成overfitting现象。</p>
<h3 id="正则化解决过拟合问题"><a href="#正则化解决过拟合问题" class="headerlink" title="正则化解决过拟合问题"></a>正则化解决过拟合问题</h3><p>Regularization可以使得曲线变得更加平滑，使得training data 上的error变大，testing data上的error变小。</p>
<p>在之前的Loss function中我们只考虑了预测值的error，即</p>
<script type="math/tex; mode=display">
L(f) = \Sigma(\widehat{y}^n-(b+w_n*Xcp^n))</script><p>而Regularization则是在这个基础上加上λ ∑ ( w i ) 2 ，即</p>
<script type="math/tex; mode=display">
L(f) = \Sigma(\widehat{y}^n-(b+w_n*Xcp^n)) + \lambda\Sigma(w_i)^2</script><p>而平滑的意思就是wi要尽可能的小即趋向于0，为什么呢？</p>
<p>因为当我们的function对input越不敏感，说明这个function越平滑，当我们的参数wi趋向于0时，output对input的输入是不敏感的。而我们不考虑把bais放进入的原因是bais与function的平滑度没有关系，它的作用只是把function上下移动而已。</p>
<p>那为什么我们喜欢平滑的function呢？</p>
<p>因为当我们对input不敏感时，一些noise噪声对我们的预测值影响不算很大，会给一个比较好的结果。</p>
<p>而λ则需要我们手动去调整以取得最好的值，当λ设定的越大，我们找到的function就越平滑。</p>
<p>观察下图可知，当我们的λ越大的时候，在training data上得到的error其实是越大的，但是这件事情是非常合理的，因为当λ越大的时候，我们就越倾向于考虑w的值而越少考虑error的大小；但是有趣的是，虽然在training data上得到的error越大，但是在testing data上得到的error可能会是比较小的。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923232300335.png" alt="image-20220923232300335" style="zoom:80%;"></p>
<p>我们喜欢比较平滑的function，因为它对noise不那么sensitive；但是我们又不喜欢太平滑的function，因为它就失去了对data拟合的能力；而function的平滑程度，就需要通过调整λ来决定。</p>
<h1 id="Error"><a href="#Error" class="headerlink" title="Error"></a>Error</h1><h2 id="error的来源"><a href="#error的来源" class="headerlink" title="error的来源"></a>error的来源</h2><p>error from testing data 来源于两个方向：bais和variance</p>
<p>通常我们可以通过一堆training data分组来确定多个$ f^<em> $,对这些$ f^</em> $取平均我们可以得到$\overline{f}$</p>
<p>由下图我们可以观测到，如果bais造成的误差越大，那么我们定义的function离真正正确的model就越远，而variance造成的误差越大，说明我们确定的多个$ f^* $越离散。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923234401554.png" alt="image-20220923234401554" style="zoom:80%;"></p>
<p>对上面宝可梦的例子，我们训练1000条数据，把1000条分为100组，确定不同的$ f^* $，用图像直观的展示出来。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923234821727.png" alt="image-20220923234821727" style="zoom:80%;"></p>
<p>我们可以简单地观测出一个结论：当模型越简单，方差越小，可以看到function基本都集中起来，但是简单模型往往平均起来很偏移正确的model；而当模型越复杂，方差就会随之变大，fucntion也很离散，但是他们的平均往往和正确的model很接近。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220923235447158.png" alt="image-20220923235447158" style="zoom:80%;"></p>
<h2 id="解决error的方法"><a href="#解决error的方法" class="headerlink" title="解决error的方法"></a>解决error的方法</h2><p>一般情况下，我们会出现两种问题：</p>
<ul>
<li>如果你的model并不能fit训练集，那这个model有很大的bais——欠拟合问题</li>
<li>如果你的model能fit训练集，但是在测试集上有很大的error,那model有很大的variance——过拟合问题</li>
</ul>
<p>对于欠拟合问题，我们解决方法通常有两种：</p>
<ol>
<li>引入更多的object的属性变量</li>
<li>重新定义一个更复杂的model</li>
</ol>
<p>对于过拟合问题，我们解决方法也有两种：</p>
<ol>
<li>收集更多的training data</li>
<li>正则化解决过拟合问题</li>
</ol>
<p>在实际的选择模型中，我们在往往采用n重交叉验证的方式，这样得到的function往往具有良好的泛化能力。</p>
<p><img src="https://pluto-1300780100.cos.ap-nanjing.myqcloud.com/img/image-20220924000734113.png" alt="image-20220924000734113" style="zoom:80%;"></p>
<h1 id="结语"><a href="#结语" class="headerlink" title="结语"></a>结语</h1><p>骚话想不出来了，老了呀😭</p>

                
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